why would this

(1 i)^T * (e ^ (-t/2)) * (cos(t) + i*sin(t))

equal this

( e^(-t/2)*cos(t)  -e^(-t/2)sin(t) )^T + i*( e^(-t/2)*sin(t)   e^(-t/2)*cos(t) )^T

note that ^T designates the transpose of the matrix. this is from
Boyce, DiPrima, "Elementary Differential Equations and Boundary Value
Problems" 8ed, p.403, section 7.6 "complex eigenvalues" example 1.
note that this is not a homework exercise, it is an example in the
section that just isn't explained well. if you need some sense of
where I'm coming from, it is from this class:
http://math.berkeley.edu/~rieffel/54web.html

thanks! I bet it is just some identity that I didn't think of.

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Clarification of Question by ralphs-ga on 19 Dec 2004 06:57 PST

I think I may have figured it out...update soon

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Request for Question Clarification by mathtalk-ga on 19 Dec 2004 07:42 PST

Hi, ralphs-ga:

It's difficult to parse your expression without some hints beyond ^T
being a transpose of a matrix.

Perhaps (1 i)^T denotes a row vector times this matrix transpose.

If so, the big question is whether:

(e ^ (-t/2)) * (cos(t) + i*sin(t))

is simply a (complex) scalar.  Scalar multiplication commutes with
matrix multiplication.  You will probably work it out from there.

regards, mathtalk-ga